On the business side, a big plus of Ajax is that your user can access the app with absolutely zero effort, oncew they've got hold of the URL. The flipside of this is that they have made no investment in using it, and will drop it and move on if it isn't compelling. That could be an advantage or a disadvantage depending on how you look at it
I think the main disadvantage is the expected end-user experience. People are conditioned to expect thick clients to behave one way, and Web applications to behave another. AJAX is nothing more than describing an approach to achieve the same effect of a thick client via a thin client. While I may seem dismissive about AJAX - which I am - I also recognize that it is now a part of our parlance, and that the chief hurdle is in helping the end-user to understand the principle of AJAX + Web = Thick Client mentality.
I think the main disadvantage is the expected end-user experience.
I think this might be true for poor implementations of Ajax. But if used correctly, Ajax should enhance the user experience. In most cases, these enhancements don't even need to be all that noticeable to the user. Examples of a good Ajax implementation would be refreshing part of the page instead of the whole thing. A bad implementation would be a type-ahead feature that slows the data entry process down rather than speeding it up.
So in my mind, the disadvantages of Ajax are the same as those of any web technology... it needs to be implemented judiciously, with the user's experience in mind.
Failures are practice shoots for success.